Introduction to Auctions

A short history of auctions

• 500 B.C. Babylon: women were auctioned off as wives (Herodot)

• ~150: Roman soldiers sold war plunder at auction.

• 1744: Sotheby's was created and Christie's in 1766. (mainly art)

• 1887: Sale of fruits and vegetables in the Netherlands through auctions

• 1995: eBay was launched

• 2000/01: UMTS - G3 auction for radio spectrum in Europe (revenue of >50bn EUR)

Use of auctions today

• Treasury bills

• Drilling rights on oil fields

• Privatisation of firms and other assets

• Takeover battles

• Procurement

• and many more

An auction is a market institution with an explicit set of rules determining resource allocation and prices on the basis of bids from the market participants.(McAfee, McMillan, 1987) e.g. used for

• Price Setting - difficult to determine the highest willingness to pay, however lower search cost for customer

• Lottery: not efficient since not necessarily the participant with the highest value for the good wins

• Contract awarding (Beauty Contest): might easily yield a political outcome and might lower the chance of getting new entrants to the market.

Why selling through auctions

Four basic types of auctions

• Ascending bid (English) auction the price is successively raised until only one bidder remains, and that bidder wins at the final price.

• Descending bid (Dutch) auction the price is lowered continuously; the first bidder calling out wins the object at this price.

• First Price sealed bid auction each bidder independently submits a single bid, without seeing others’ bids, the object is sold to the bidder who makes the highest bid; at a price of his bid.

• Second Price sealed bid (Vickrey) auction each bidder independently submits a single bid, without seeing others’ bids, the object is sold to the bidder with the highest bid at the second highest bid.

Strategical equivalent auctions I

• Ascending bid (English) auction the price is successively raised until only one bidder remains, and that bidder wins at the final price.

• Descending bid (Dutch) auction the price is lowered continuously; the first bidder calling out wins the object at this price.

• First Price sealed bid auction each bidder independently submits a single bid, without seeing others’ bids, the object is sold to the bidder who makes the highest bid; at a price of his bid.

• Second Price sealed bid (Vickrey) auction each bidder independently submits a single bid, without seeing others’ bids, the object is sold to the bidder with the highest bid at the second highest bid.

Strategical equivalent auctions II

• Ascending bid (English) auction the price is successively raised until only one bidder remains, and that bidder wins at the final price.

• Descending bid (Dutch) auction the price is lowered continuously; the first bidder calling out wins the object at this price.

• First Price sealed bid auction each bidder independently submits a single bid, without seeing others’ bids, the object is sold to the bidder who makes the highest bid; at a price of his bid.

• Second Price sealed bid (Vickrey) auction each bidder independently submits a single bid, without seeing others’ bids, the object is sold to the bidder with the highest bid at the second highest bid.

Strategic under-bidding(three cases)

Strategic over-bidding(three cases)

my bid opponents bid my profitmy loss

Truth telling a dominant strategySecond Price Auctions

v

price

Truth Telling is the dominant strategy for Second Price auctions => bid shading is not a good (dominant) strategy.

What happens in a First Price auction? Truce telling can not be a optimal strategy since:

for a bid b=vi the expected revenue is 0; E[r]=0

Therefore the optimal bid b* must be smaller than bidders i true value vi => bid shading is a good (dominant) strategy.

Truth telling a dominant strategy

Recall the first price auction: all players state a bid, and the winner is the player with maximum bid, and has to pay his bid as price.

Assumptions:- there are n>=2 bidders- every bidder draws (independently) a valuation v from [0,1]- if bidder i submits the highest bid b* she realizes a economic gain of vi-b*

Claim: best strategy for bidder with value vi is to bid:

Proof: probability of winning for bid b is: and the expected value:

First derivative with respect to b (set equal to 0 to find the maximum) gives:

which reduces to:

Strategy for first price auction

vi

(n !1)

n

bn

n !1

"

# $

%

& '

n!1

(vi! b)

bn

n !1

"

# $

%

& '

n!1

!bn

n !1

"

# $

%

& '

n!1

+ (vi! b)(n !1)bn!2

n

n !1

"

# $

%

& '

n!1

= 0

!b + (vi! b)(n !1) = 0

b = vi

(n !1)

n

Strategy for first price auction

In a First price auction the dominant strategy is given by:

optimal bid is in case of n=2

According to the Revenue Equivalence Theorem (here without proof): Second and First Price auctions generate the same revenue for bidder and auctioneer. If so, why bother about auction design at all?

b = vi

(n !1)

n

bi

=1

2vi

• Available time to sell the goods (fish, flowers, tobacco) =>Dutch auction

• Risk of collusion (limited amount of buyers) => First Price auction

• Predatory behavior (a strong bidder participating might discourage new entrants) => Sealed bid auction

Auction design needs to consider

v

price

First Price Auction

profit

my bid opponents bid my profitmy loss

Collusion

lossSecond Price Auction

agreed priceagreed price

• Bidding on drilling rights on oil fields

• All bidders form their estimate value vi

according to their own geological evaluation

• Each bidder makes a single bid

• If you win you probably overestimated the true value of the oil field

The Winners Curse

Interdependent valuation

• All bidders are risk neutral

• private values are independent drawn from a uniform distribution

Underlying assumptions

The marriage market